VS 265
Linear timeinvariant systems and convolution
Linear, timeinvariant systems
 Let us consider a dynamical system with input
and output
:
 Such a system is said to be a linear, timeinvariant
system if it obeys the laws of superposition and scaling over time. That
is, if you observe an output signal
in response to an input signal
, and you later observe an output
in response to input
, then the response to the combination
is just
.
 One way to characterize a linear, timeinvariant system
is by measuring its impulse response function. This is the response
you would obtain to a small pulse of unit amplitude,
, where
Let’s call the output signal we measure in response to such an input
.
 Once we have measured the impulse response function,
, we can compute the output
in response to any complex input
. The first step is to consider the signal
to be composed of a superposition of unit impulses of different amplitudes:
Now since the system is also timeinvariant (i.e., it does not change
its behavior over time), then we know that the response to a shifted impulse
is just
. And so the response to a weighted sum of such shifted impulses is just
a weighted sum of the resulting shifted impulse response functions. Or in
the language of mathematics:
 Thus, the output
in response to the input signal
may be written as
(1)
And in the limit that the spacing between time samples becomes infinitesimally
small, this relation becomes exact and the sum turns into an integral:
(2)
Convolution
 The expression above is known as the convolution sum
(1) or convolution integral (2). It tells us how to predict the output
of a linear, timeinvariant system in response to any arbitrary input signal.
 The mathematical shorthand notation for the convolution
operation is to use the
symbol as follows:
 One way of interpreting the convolution sum is just as
we developed it above  i.e., it is simply a linear superposition of impulse
response functions
each of which is multiplied by
.
 The other (more common way) of interpreting the convolution
sum is that it tells us that the output is computed by taking a weighted
sum of the present and past input values. We can see this by writing
out the sum in (1) above:
where we have assumed here for now that the times
are spaced by one unit of time. Note also that we do not sum over values
of
for which
. The reason is that for any physical system,
is defined only for
. This indeed makes sense, because otherwise we would need to know future
values of the input in order to compute the present output.

Typically the impulse response function decays away with time,
and there is a point at which we can consider it to be essentially zero and
so we can truncate the expansion above at a certain number of “taps” (discrete
samples).
 The convolution operation may also be thought of as a
filtering operation on the signal
, where the impulse response function
is acting as the filter. The shape of
determines which properties of the original signal
are “filtered out.” The design of filters is usually best thought of in
the frequency domain, which we turn to next....